# Properties

 Label 4205.2.a.r.1.1 Level $4205$ Weight $2$ Character 4205.1 Self dual yes Analytic conductor $33.577$ Analytic rank $1$ Dimension $12$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4205,2,Mod(1,4205)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4205 = 5 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4205.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$33.5770940499$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{11} - 14 x^{10} + 11 x^{9} + 72 x^{8} - 41 x^{7} - 164 x^{6} + 62 x^{5} + 156 x^{4} + \cdots - 1$$ x^12 - x^11 - 14*x^10 + 11*x^9 + 72*x^8 - 41*x^7 - 164*x^6 + 62*x^5 + 156*x^4 - 43*x^3 - 46*x^2 + 15*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.15738$$ of defining polynomial Character $$\chi$$ $$=$$ 4205.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.15738 q^{2} -0.688118 q^{3} +2.65427 q^{4} -1.00000 q^{5} +1.48453 q^{6} +3.16489 q^{7} -1.41152 q^{8} -2.52649 q^{9} +O(q^{10})$$ $$q-2.15738 q^{2} -0.688118 q^{3} +2.65427 q^{4} -1.00000 q^{5} +1.48453 q^{6} +3.16489 q^{7} -1.41152 q^{8} -2.52649 q^{9} +2.15738 q^{10} +1.52441 q^{11} -1.82645 q^{12} -5.35291 q^{13} -6.82785 q^{14} +0.688118 q^{15} -2.26338 q^{16} +4.43761 q^{17} +5.45060 q^{18} +1.07517 q^{19} -2.65427 q^{20} -2.17781 q^{21} -3.28872 q^{22} -4.89796 q^{23} +0.971290 q^{24} +1.00000 q^{25} +11.5482 q^{26} +3.80288 q^{27} +8.40047 q^{28} -1.48453 q^{30} +6.65314 q^{31} +7.70599 q^{32} -1.04897 q^{33} -9.57359 q^{34} -3.16489 q^{35} -6.70601 q^{36} +5.72046 q^{37} -2.31955 q^{38} +3.68343 q^{39} +1.41152 q^{40} -4.10840 q^{41} +4.69837 q^{42} -10.4700 q^{43} +4.04619 q^{44} +2.52649 q^{45} +10.5668 q^{46} +0.173845 q^{47} +1.55747 q^{48} +3.01650 q^{49} -2.15738 q^{50} -3.05360 q^{51} -14.2081 q^{52} +7.21860 q^{53} -8.20424 q^{54} -1.52441 q^{55} -4.46729 q^{56} -0.739844 q^{57} -2.16431 q^{59} +1.82645 q^{60} -9.58307 q^{61} -14.3533 q^{62} -7.99606 q^{63} -12.0980 q^{64} +5.35291 q^{65} +2.26303 q^{66} -7.82287 q^{67} +11.7786 q^{68} +3.37038 q^{69} +6.82785 q^{70} +3.96962 q^{71} +3.56619 q^{72} -12.9518 q^{73} -12.3412 q^{74} -0.688118 q^{75} +2.85380 q^{76} +4.82457 q^{77} -7.94656 q^{78} +0.565815 q^{79} +2.26338 q^{80} +4.96265 q^{81} +8.86337 q^{82} +10.3651 q^{83} -5.78052 q^{84} -4.43761 q^{85} +22.5877 q^{86} -2.15172 q^{88} -3.54640 q^{89} -5.45060 q^{90} -16.9413 q^{91} -13.0005 q^{92} -4.57814 q^{93} -0.375049 q^{94} -1.07517 q^{95} -5.30263 q^{96} +12.7595 q^{97} -6.50773 q^{98} -3.85140 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + q^{2} - q^{3} + 5 q^{4} - 12 q^{5} - 9 q^{6} - 5 q^{7} + 6 q^{8} + 7 q^{9}+O(q^{10})$$ 12 * q + q^2 - q^3 + 5 * q^4 - 12 * q^5 - 9 * q^6 - 5 * q^7 + 6 * q^8 + 7 * q^9 $$12 q + q^{2} - q^{3} + 5 q^{4} - 12 q^{5} - 9 q^{6} - 5 q^{7} + 6 q^{8} + 7 q^{9} - q^{10} + q^{11} - q^{12} - 5 q^{13} - 5 q^{14} + q^{15} - 9 q^{16} + 20 q^{17} - 5 q^{18} - 8 q^{19} - 5 q^{20} + 3 q^{22} - 8 q^{23} + 10 q^{24} + 12 q^{25} - 17 q^{26} - 25 q^{27} + 10 q^{28} + 9 q^{30} - 7 q^{31} + 3 q^{32} + q^{33} - 8 q^{34} + 5 q^{35} - 8 q^{36} + 14 q^{38} + 29 q^{39} - 6 q^{40} - 4 q^{41} + 13 q^{42} - 15 q^{43} + 2 q^{44} - 7 q^{45} + 24 q^{46} - 39 q^{47} + 2 q^{48} - 19 q^{49} + q^{50} - 32 q^{51} - 32 q^{52} + 12 q^{53} - 34 q^{54} - q^{55} - 19 q^{56} + 10 q^{57} - 19 q^{59} + q^{60} + 28 q^{61} - 13 q^{62} - 40 q^{63} - 34 q^{64} + 5 q^{65} + 48 q^{66} - 38 q^{67} + 18 q^{68} + 18 q^{69} + 5 q^{70} - 20 q^{71} - 6 q^{72} - 5 q^{73} - 12 q^{74} - q^{75} - 19 q^{76} + 32 q^{77} - 14 q^{78} + 13 q^{79} + 9 q^{80} - 32 q^{81} - 34 q^{82} + 15 q^{83} - 32 q^{84} - 20 q^{85} - 9 q^{86} - 10 q^{88} - 22 q^{89} + 5 q^{90} - 46 q^{91} - 31 q^{92} - 6 q^{93} - 13 q^{94} + 8 q^{95} - 3 q^{96} + 53 q^{97} - 8 q^{98} - 44 q^{99}+O(q^{100})$$ 12 * q + q^2 - q^3 + 5 * q^4 - 12 * q^5 - 9 * q^6 - 5 * q^7 + 6 * q^8 + 7 * q^9 - q^10 + q^11 - q^12 - 5 * q^13 - 5 * q^14 + q^15 - 9 * q^16 + 20 * q^17 - 5 * q^18 - 8 * q^19 - 5 * q^20 + 3 * q^22 - 8 * q^23 + 10 * q^24 + 12 * q^25 - 17 * q^26 - 25 * q^27 + 10 * q^28 + 9 * q^30 - 7 * q^31 + 3 * q^32 + q^33 - 8 * q^34 + 5 * q^35 - 8 * q^36 + 14 * q^38 + 29 * q^39 - 6 * q^40 - 4 * q^41 + 13 * q^42 - 15 * q^43 + 2 * q^44 - 7 * q^45 + 24 * q^46 - 39 * q^47 + 2 * q^48 - 19 * q^49 + q^50 - 32 * q^51 - 32 * q^52 + 12 * q^53 - 34 * q^54 - q^55 - 19 * q^56 + 10 * q^57 - 19 * q^59 + q^60 + 28 * q^61 - 13 * q^62 - 40 * q^63 - 34 * q^64 + 5 * q^65 + 48 * q^66 - 38 * q^67 + 18 * q^68 + 18 * q^69 + 5 * q^70 - 20 * q^71 - 6 * q^72 - 5 * q^73 - 12 * q^74 - q^75 - 19 * q^76 + 32 * q^77 - 14 * q^78 + 13 * q^79 + 9 * q^80 - 32 * q^81 - 34 * q^82 + 15 * q^83 - 32 * q^84 - 20 * q^85 - 9 * q^86 - 10 * q^88 - 22 * q^89 + 5 * q^90 - 46 * q^91 - 31 * q^92 - 6 * q^93 - 13 * q^94 + 8 * q^95 - 3 * q^96 + 53 * q^97 - 8 * q^98 - 44 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.15738 −1.52550 −0.762748 0.646696i $$-0.776150\pi$$
−0.762748 + 0.646696i $$0.776150\pi$$
$$3$$ −0.688118 −0.397285 −0.198643 0.980072i $$-0.563653\pi$$
−0.198643 + 0.980072i $$0.563653\pi$$
$$4$$ 2.65427 1.32714
$$5$$ −1.00000 −0.447214
$$6$$ 1.48453 0.606057
$$7$$ 3.16489 1.19621 0.598107 0.801416i $$-0.295920\pi$$
0.598107 + 0.801416i $$0.295920\pi$$
$$8$$ −1.41152 −0.499046
$$9$$ −2.52649 −0.842165
$$10$$ 2.15738 0.682222
$$11$$ 1.52441 0.459626 0.229813 0.973235i $$-0.426189\pi$$
0.229813 + 0.973235i $$0.426189\pi$$
$$12$$ −1.82645 −0.527252
$$13$$ −5.35291 −1.48463 −0.742315 0.670051i $$-0.766272\pi$$
−0.742315 + 0.670051i $$0.766272\pi$$
$$14$$ −6.82785 −1.82482
$$15$$ 0.688118 0.177671
$$16$$ −2.26338 −0.565844
$$17$$ 4.43761 1.07628 0.538139 0.842856i $$-0.319128\pi$$
0.538139 + 0.842856i $$0.319128\pi$$
$$18$$ 5.45060 1.28472
$$19$$ 1.07517 0.246661 0.123331 0.992366i $$-0.460642\pi$$
0.123331 + 0.992366i $$0.460642\pi$$
$$20$$ −2.65427 −0.593514
$$21$$ −2.17781 −0.475238
$$22$$ −3.28872 −0.701157
$$23$$ −4.89796 −1.02130 −0.510648 0.859790i $$-0.670594\pi$$
−0.510648 + 0.859790i $$0.670594\pi$$
$$24$$ 0.971290 0.198264
$$25$$ 1.00000 0.200000
$$26$$ 11.5482 2.26480
$$27$$ 3.80288 0.731865
$$28$$ 8.40047 1.58754
$$29$$ 0 0
$$30$$ −1.48453 −0.271037
$$31$$ 6.65314 1.19494 0.597469 0.801892i $$-0.296173\pi$$
0.597469 + 0.801892i $$0.296173\pi$$
$$32$$ 7.70599 1.36224
$$33$$ −1.04897 −0.182603
$$34$$ −9.57359 −1.64186
$$35$$ −3.16489 −0.534963
$$36$$ −6.70601 −1.11767
$$37$$ 5.72046 0.940437 0.470219 0.882550i $$-0.344175\pi$$
0.470219 + 0.882550i $$0.344175\pi$$
$$38$$ −2.31955 −0.376280
$$39$$ 3.68343 0.589822
$$40$$ 1.41152 0.223180
$$41$$ −4.10840 −0.641625 −0.320812 0.947143i $$-0.603956\pi$$
−0.320812 + 0.947143i $$0.603956\pi$$
$$42$$ 4.69837 0.724974
$$43$$ −10.4700 −1.59666 −0.798330 0.602220i $$-0.794283\pi$$
−0.798330 + 0.602220i $$0.794283\pi$$
$$44$$ 4.04619 0.609987
$$45$$ 2.52649 0.376627
$$46$$ 10.5668 1.55798
$$47$$ 0.173845 0.0253579 0.0126789 0.999920i $$-0.495964\pi$$
0.0126789 + 0.999920i $$0.495964\pi$$
$$48$$ 1.55747 0.224802
$$49$$ 3.01650 0.430929
$$50$$ −2.15738 −0.305099
$$51$$ −3.05360 −0.427589
$$52$$ −14.2081 −1.97031
$$53$$ 7.21860 0.991551 0.495775 0.868451i $$-0.334884\pi$$
0.495775 + 0.868451i $$0.334884\pi$$
$$54$$ −8.20424 −1.11646
$$55$$ −1.52441 −0.205551
$$56$$ −4.46729 −0.596966
$$57$$ −0.739844 −0.0979948
$$58$$ 0 0
$$59$$ −2.16431 −0.281769 −0.140884 0.990026i $$-0.544995\pi$$
−0.140884 + 0.990026i $$0.544995\pi$$
$$60$$ 1.82645 0.235794
$$61$$ −9.58307 −1.22699 −0.613493 0.789700i $$-0.710236\pi$$
−0.613493 + 0.789700i $$0.710236\pi$$
$$62$$ −14.3533 −1.82287
$$63$$ −7.99606 −1.00741
$$64$$ −12.0980 −1.51225
$$65$$ 5.35291 0.663947
$$66$$ 2.26303 0.278559
$$67$$ −7.82287 −0.955716 −0.477858 0.878437i $$-0.658587\pi$$
−0.477858 + 0.878437i $$0.658587\pi$$
$$68$$ 11.7786 1.42837
$$69$$ 3.37038 0.405746
$$70$$ 6.82785 0.816084
$$71$$ 3.96962 0.471108 0.235554 0.971861i $$-0.424310\pi$$
0.235554 + 0.971861i $$0.424310\pi$$
$$72$$ 3.56619 0.420279
$$73$$ −12.9518 −1.51589 −0.757947 0.652316i $$-0.773798\pi$$
−0.757947 + 0.652316i $$0.773798\pi$$
$$74$$ −12.3412 −1.43463
$$75$$ −0.688118 −0.0794570
$$76$$ 2.85380 0.327353
$$77$$ 4.82457 0.549811
$$78$$ −7.94656 −0.899770
$$79$$ 0.565815 0.0636591 0.0318296 0.999493i $$-0.489867\pi$$
0.0318296 + 0.999493i $$0.489867\pi$$
$$80$$ 2.26338 0.253053
$$81$$ 4.96265 0.551406
$$82$$ 8.86337 0.978796
$$83$$ 10.3651 1.13772 0.568859 0.822435i $$-0.307385\pi$$
0.568859 + 0.822435i $$0.307385\pi$$
$$84$$ −5.78052 −0.630706
$$85$$ −4.43761 −0.481326
$$86$$ 22.5877 2.43570
$$87$$ 0 0
$$88$$ −2.15172 −0.229375
$$89$$ −3.54640 −0.375917 −0.187959 0.982177i $$-0.560187\pi$$
−0.187959 + 0.982177i $$0.560187\pi$$
$$90$$ −5.45060 −0.574544
$$91$$ −16.9413 −1.77594
$$92$$ −13.0005 −1.35540
$$93$$ −4.57814 −0.474731
$$94$$ −0.375049 −0.0386834
$$95$$ −1.07517 −0.110310
$$96$$ −5.30263 −0.541197
$$97$$ 12.7595 1.29553 0.647765 0.761840i $$-0.275704\pi$$
0.647765 + 0.761840i $$0.275704\pi$$
$$98$$ −6.50773 −0.657380
$$99$$ −3.85140 −0.387081
$$100$$ 2.65427 0.265427
$$101$$ −8.43874 −0.839686 −0.419843 0.907597i $$-0.637915\pi$$
−0.419843 + 0.907597i $$0.637915\pi$$
$$102$$ 6.58776 0.652285
$$103$$ 17.2403 1.69874 0.849370 0.527798i $$-0.176982\pi$$
0.849370 + 0.527798i $$0.176982\pi$$
$$104$$ 7.55572 0.740899
$$105$$ 2.17781 0.212533
$$106$$ −15.5732 −1.51261
$$107$$ 5.53818 0.535396 0.267698 0.963503i $$-0.413737\pi$$
0.267698 + 0.963503i $$0.413737\pi$$
$$108$$ 10.0939 0.971285
$$109$$ −13.8570 −1.32726 −0.663631 0.748060i $$-0.730986\pi$$
−0.663631 + 0.748060i $$0.730986\pi$$
$$110$$ 3.28872 0.313567
$$111$$ −3.93635 −0.373622
$$112$$ −7.16333 −0.676871
$$113$$ −6.36180 −0.598468 −0.299234 0.954180i $$-0.596731\pi$$
−0.299234 + 0.954180i $$0.596731\pi$$
$$114$$ 1.59612 0.149491
$$115$$ 4.89796 0.456738
$$116$$ 0 0
$$117$$ 13.5241 1.25030
$$118$$ 4.66923 0.429837
$$119$$ 14.0445 1.28746
$$120$$ −0.971290 −0.0886662
$$121$$ −8.67618 −0.788744
$$122$$ 20.6743 1.87176
$$123$$ 2.82707 0.254908
$$124$$ 17.6592 1.58585
$$125$$ −1.00000 −0.0894427
$$126$$ 17.2505 1.53680
$$127$$ −1.57064 −0.139372 −0.0696861 0.997569i $$-0.522200\pi$$
−0.0696861 + 0.997569i $$0.522200\pi$$
$$128$$ 10.6879 0.944685
$$129$$ 7.20460 0.634330
$$130$$ −11.5482 −1.01285
$$131$$ 17.2730 1.50915 0.754574 0.656215i $$-0.227843\pi$$
0.754574 + 0.656215i $$0.227843\pi$$
$$132$$ −2.78426 −0.242339
$$133$$ 3.40279 0.295060
$$134$$ 16.8769 1.45794
$$135$$ −3.80288 −0.327300
$$136$$ −6.26375 −0.537112
$$137$$ 17.3973 1.48635 0.743177 0.669094i $$-0.233318\pi$$
0.743177 + 0.669094i $$0.233318\pi$$
$$138$$ −7.27117 −0.618963
$$139$$ −20.1880 −1.71232 −0.856161 0.516709i $$-0.827157\pi$$
−0.856161 + 0.516709i $$0.827157\pi$$
$$140$$ −8.40047 −0.709970
$$141$$ −0.119626 −0.0100743
$$142$$ −8.56398 −0.718673
$$143$$ −8.16001 −0.682375
$$144$$ 5.71841 0.476534
$$145$$ 0 0
$$146$$ 27.9419 2.31249
$$147$$ −2.07571 −0.171202
$$148$$ 15.1837 1.24809
$$149$$ −10.9422 −0.896419 −0.448209 0.893929i $$-0.647938\pi$$
−0.448209 + 0.893929i $$0.647938\pi$$
$$150$$ 1.48453 0.121211
$$151$$ 11.3113 0.920504 0.460252 0.887788i $$-0.347759\pi$$
0.460252 + 0.887788i $$0.347759\pi$$
$$152$$ −1.51762 −0.123095
$$153$$ −11.2116 −0.906403
$$154$$ −10.4084 −0.838735
$$155$$ −6.65314 −0.534393
$$156$$ 9.77684 0.782774
$$157$$ 4.26189 0.340136 0.170068 0.985432i $$-0.445601\pi$$
0.170068 + 0.985432i $$0.445601\pi$$
$$158$$ −1.22068 −0.0971117
$$159$$ −4.96725 −0.393928
$$160$$ −7.70599 −0.609212
$$161$$ −15.5015 −1.22169
$$162$$ −10.7063 −0.841167
$$163$$ 21.4615 1.68099 0.840497 0.541816i $$-0.182263\pi$$
0.840497 + 0.541816i $$0.182263\pi$$
$$164$$ −10.9048 −0.851524
$$165$$ 1.04897 0.0816623
$$166$$ −22.3614 −1.73558
$$167$$ 12.4882 0.966369 0.483185 0.875519i $$-0.339480\pi$$
0.483185 + 0.875519i $$0.339480\pi$$
$$168$$ 3.07402 0.237166
$$169$$ 15.6537 1.20413
$$170$$ 9.57359 0.734261
$$171$$ −2.71641 −0.207729
$$172$$ −27.7903 −2.11899
$$173$$ −11.6421 −0.885135 −0.442568 0.896735i $$-0.645932\pi$$
−0.442568 + 0.896735i $$0.645932\pi$$
$$174$$ 0 0
$$175$$ 3.16489 0.239243
$$176$$ −3.45031 −0.260077
$$177$$ 1.48930 0.111943
$$178$$ 7.65092 0.573460
$$179$$ 18.2083 1.36095 0.680477 0.732769i $$-0.261772\pi$$
0.680477 + 0.732769i $$0.261772\pi$$
$$180$$ 6.70601 0.499836
$$181$$ −20.6043 −1.53151 −0.765753 0.643134i $$-0.777634\pi$$
−0.765753 + 0.643134i $$0.777634\pi$$
$$182$$ 36.5489 2.70918
$$183$$ 6.59428 0.487463
$$184$$ 6.91355 0.509674
$$185$$ −5.72046 −0.420576
$$186$$ 9.87678 0.724201
$$187$$ 6.76472 0.494685
$$188$$ 0.461432 0.0336534
$$189$$ 12.0357 0.875467
$$190$$ 2.31955 0.168278
$$191$$ −14.4485 −1.04546 −0.522729 0.852499i $$-0.675086\pi$$
−0.522729 + 0.852499i $$0.675086\pi$$
$$192$$ 8.32483 0.600793
$$193$$ 10.0650 0.724498 0.362249 0.932081i $$-0.382009\pi$$
0.362249 + 0.932081i $$0.382009\pi$$
$$194$$ −27.5270 −1.97633
$$195$$ −3.68343 −0.263776
$$196$$ 8.00662 0.571902
$$197$$ −16.1658 −1.15177 −0.575884 0.817532i $$-0.695342\pi$$
−0.575884 + 0.817532i $$0.695342\pi$$
$$198$$ 8.30893 0.590490
$$199$$ 3.08996 0.219042 0.109521 0.993984i $$-0.465068\pi$$
0.109521 + 0.993984i $$0.465068\pi$$
$$200$$ −1.41152 −0.0998092
$$201$$ 5.38306 0.379692
$$202$$ 18.2055 1.28094
$$203$$ 0 0
$$204$$ −8.10508 −0.567469
$$205$$ 4.10840 0.286943
$$206$$ −37.1939 −2.59142
$$207$$ 12.3747 0.860099
$$208$$ 12.1157 0.840069
$$209$$ 1.63900 0.113372
$$210$$ −4.69837 −0.324218
$$211$$ −19.8418 −1.36597 −0.682983 0.730434i $$-0.739318\pi$$
−0.682983 + 0.730434i $$0.739318\pi$$
$$212$$ 19.1601 1.31592
$$213$$ −2.73157 −0.187164
$$214$$ −11.9479 −0.816744
$$215$$ 10.4700 0.714048
$$216$$ −5.36783 −0.365234
$$217$$ 21.0564 1.42940
$$218$$ 29.8948 2.02473
$$219$$ 8.91237 0.602242
$$220$$ −4.04619 −0.272794
$$221$$ −23.7541 −1.59787
$$222$$ 8.49219 0.569958
$$223$$ −10.5744 −0.708115 −0.354058 0.935224i $$-0.615198\pi$$
−0.354058 + 0.935224i $$0.615198\pi$$
$$224$$ 24.3886 1.62953
$$225$$ −2.52649 −0.168433
$$226$$ 13.7248 0.912960
$$227$$ 5.96180 0.395698 0.197849 0.980232i $$-0.436604\pi$$
0.197849 + 0.980232i $$0.436604\pi$$
$$228$$ −1.96375 −0.130053
$$229$$ −2.39894 −0.158526 −0.0792631 0.996854i $$-0.525257\pi$$
−0.0792631 + 0.996854i $$0.525257\pi$$
$$230$$ −10.5668 −0.696751
$$231$$ −3.31988 −0.218432
$$232$$ 0 0
$$233$$ 14.0222 0.918624 0.459312 0.888275i $$-0.348096\pi$$
0.459312 + 0.888275i $$0.348096\pi$$
$$234$$ −29.1766 −1.90733
$$235$$ −0.173845 −0.0113404
$$236$$ −5.74467 −0.373946
$$237$$ −0.389347 −0.0252908
$$238$$ −30.2993 −1.96401
$$239$$ −10.6248 −0.687263 −0.343632 0.939105i $$-0.611657\pi$$
−0.343632 + 0.939105i $$0.611657\pi$$
$$240$$ −1.55747 −0.100534
$$241$$ −22.1358 −1.42589 −0.712947 0.701218i $$-0.752640\pi$$
−0.712947 + 0.701218i $$0.752640\pi$$
$$242$$ 18.7178 1.20323
$$243$$ −14.8235 −0.950930
$$244$$ −25.4361 −1.62838
$$245$$ −3.01650 −0.192717
$$246$$ −6.09905 −0.388861
$$247$$ −5.75529 −0.366201
$$248$$ −9.39101 −0.596330
$$249$$ −7.13242 −0.451999
$$250$$ 2.15738 0.136444
$$251$$ 8.87412 0.560129 0.280065 0.959981i $$-0.409644\pi$$
0.280065 + 0.959981i $$0.409644\pi$$
$$252$$ −21.2237 −1.33697
$$253$$ −7.46649 −0.469414
$$254$$ 3.38847 0.212612
$$255$$ 3.05360 0.191224
$$256$$ 1.13812 0.0711326
$$257$$ −20.0115 −1.24828 −0.624140 0.781312i $$-0.714551\pi$$
−0.624140 + 0.781312i $$0.714551\pi$$
$$258$$ −15.5430 −0.967667
$$259$$ 18.1046 1.12496
$$260$$ 14.2081 0.881148
$$261$$ 0 0
$$262$$ −37.2644 −2.30220
$$263$$ 19.2314 1.18586 0.592930 0.805254i $$-0.297971\pi$$
0.592930 + 0.805254i $$0.297971\pi$$
$$264$$ 1.48064 0.0911271
$$265$$ −7.21860 −0.443435
$$266$$ −7.34111 −0.450112
$$267$$ 2.44034 0.149346
$$268$$ −20.7640 −1.26837
$$269$$ −22.2982 −1.35955 −0.679774 0.733422i $$-0.737922\pi$$
−0.679774 + 0.733422i $$0.737922\pi$$
$$270$$ 8.20424 0.499294
$$271$$ −5.09639 −0.309584 −0.154792 0.987947i $$-0.549471\pi$$
−0.154792 + 0.987947i $$0.549471\pi$$
$$272$$ −10.0440 −0.609006
$$273$$ 11.6576 0.705553
$$274$$ −37.5326 −2.26743
$$275$$ 1.52441 0.0919252
$$276$$ 8.94591 0.538480
$$277$$ −23.9387 −1.43834 −0.719168 0.694836i $$-0.755477\pi$$
−0.719168 + 0.694836i $$0.755477\pi$$
$$278$$ 43.5531 2.61214
$$279$$ −16.8091 −1.00633
$$280$$ 4.46729 0.266971
$$281$$ −1.06183 −0.0633433 −0.0316717 0.999498i $$-0.510083\pi$$
−0.0316717 + 0.999498i $$0.510083\pi$$
$$282$$ 0.258078 0.0153683
$$283$$ 9.07018 0.539166 0.269583 0.962977i $$-0.413114\pi$$
0.269583 + 0.962977i $$0.413114\pi$$
$$284$$ 10.5365 0.625225
$$285$$ 0.739844 0.0438246
$$286$$ 17.6042 1.04096
$$287$$ −13.0026 −0.767521
$$288$$ −19.4691 −1.14723
$$289$$ 2.69235 0.158374
$$290$$ 0 0
$$291$$ −8.78004 −0.514695
$$292$$ −34.3776 −2.01180
$$293$$ 15.1403 0.884504 0.442252 0.896891i $$-0.354180\pi$$
0.442252 + 0.896891i $$0.354180\pi$$
$$294$$ 4.47809 0.261167
$$295$$ 2.16431 0.126011
$$296$$ −8.07451 −0.469322
$$297$$ 5.79714 0.336384
$$298$$ 23.6064 1.36748
$$299$$ 26.2184 1.51625
$$300$$ −1.82645 −0.105450
$$301$$ −33.1364 −1.90995
$$302$$ −24.4028 −1.40422
$$303$$ 5.80685 0.333595
$$304$$ −2.43352 −0.139572
$$305$$ 9.58307 0.548725
$$306$$ 24.1876 1.38271
$$307$$ −22.9534 −1.31002 −0.655010 0.755620i $$-0.727336\pi$$
−0.655010 + 0.755620i $$0.727336\pi$$
$$308$$ 12.8057 0.729675
$$309$$ −11.8634 −0.674884
$$310$$ 14.3533 0.815214
$$311$$ −13.7927 −0.782110 −0.391055 0.920367i $$-0.627890\pi$$
−0.391055 + 0.920367i $$0.627890\pi$$
$$312$$ −5.19923 −0.294348
$$313$$ −25.8036 −1.45850 −0.729252 0.684245i $$-0.760132\pi$$
−0.729252 + 0.684245i $$0.760132\pi$$
$$314$$ −9.19451 −0.518876
$$315$$ 7.99606 0.450527
$$316$$ 1.50183 0.0844844
$$317$$ 20.4907 1.15087 0.575436 0.817847i $$-0.304832\pi$$
0.575436 + 0.817847i $$0.304832\pi$$
$$318$$ 10.7162 0.600936
$$319$$ 0 0
$$320$$ 12.0980 0.676297
$$321$$ −3.81092 −0.212705
$$322$$ 33.4426 1.86368
$$323$$ 4.77118 0.265476
$$324$$ 13.1722 0.731791
$$325$$ −5.35291 −0.296926
$$326$$ −46.3005 −2.56435
$$327$$ 9.53527 0.527302
$$328$$ 5.79908 0.320200
$$329$$ 0.550199 0.0303335
$$330$$ −2.26303 −0.124576
$$331$$ 27.3414 1.50282 0.751408 0.659837i $$-0.229375\pi$$
0.751408 + 0.659837i $$0.229375\pi$$
$$332$$ 27.5118 1.50991
$$333$$ −14.4527 −0.792003
$$334$$ −26.9418 −1.47419
$$335$$ 7.82287 0.427409
$$336$$ 4.92922 0.268911
$$337$$ −25.9403 −1.41306 −0.706529 0.707684i $$-0.749740\pi$$
−0.706529 + 0.707684i $$0.749740\pi$$
$$338$$ −33.7708 −1.83689
$$339$$ 4.37767 0.237762
$$340$$ −11.7786 −0.638786
$$341$$ 10.1421 0.549225
$$342$$ 5.86032 0.316890
$$343$$ −12.6073 −0.680731
$$344$$ 14.7786 0.796807
$$345$$ −3.37038 −0.181455
$$346$$ 25.1165 1.35027
$$347$$ −29.4588 −1.58143 −0.790715 0.612185i $$-0.790291\pi$$
−0.790715 + 0.612185i $$0.790291\pi$$
$$348$$ 0 0
$$349$$ 10.0123 0.535948 0.267974 0.963426i $$-0.413646\pi$$
0.267974 + 0.963426i $$0.413646\pi$$
$$350$$ −6.82785 −0.364964
$$351$$ −20.3565 −1.08655
$$352$$ 11.7471 0.626120
$$353$$ 20.5402 1.09325 0.546623 0.837379i $$-0.315913\pi$$
0.546623 + 0.837379i $$0.315913\pi$$
$$354$$ −3.21298 −0.170768
$$355$$ −3.96962 −0.210686
$$356$$ −9.41311 −0.498894
$$357$$ −9.66429 −0.511488
$$358$$ −39.2822 −2.07613
$$359$$ 29.9432 1.58034 0.790170 0.612888i $$-0.209992\pi$$
0.790170 + 0.612888i $$0.209992\pi$$
$$360$$ −3.56619 −0.187954
$$361$$ −17.8440 −0.939158
$$362$$ 44.4513 2.33631
$$363$$ 5.97024 0.313356
$$364$$ −44.9670 −2.35691
$$365$$ 12.9518 0.677929
$$366$$ −14.2264 −0.743623
$$367$$ 0.848599 0.0442965 0.0221483 0.999755i $$-0.492949\pi$$
0.0221483 + 0.999755i $$0.492949\pi$$
$$368$$ 11.0859 0.577895
$$369$$ 10.3799 0.540354
$$370$$ 12.3412 0.641587
$$371$$ 22.8460 1.18611
$$372$$ −12.1516 −0.630034
$$373$$ 1.49586 0.0774529 0.0387265 0.999250i $$-0.487670\pi$$
0.0387265 + 0.999250i $$0.487670\pi$$
$$374$$ −14.5940 −0.754640
$$375$$ 0.688118 0.0355343
$$376$$ −0.245385 −0.0126548
$$377$$ 0 0
$$378$$ −25.9655 −1.33552
$$379$$ 2.53234 0.130078 0.0650388 0.997883i $$-0.479283\pi$$
0.0650388 + 0.997883i $$0.479283\pi$$
$$380$$ −2.85380 −0.146397
$$381$$ 1.08079 0.0553705
$$382$$ 31.1709 1.59484
$$383$$ 2.31256 0.118166 0.0590832 0.998253i $$-0.481182\pi$$
0.0590832 + 0.998253i $$0.481182\pi$$
$$384$$ −7.35453 −0.375309
$$385$$ −4.82457 −0.245883
$$386$$ −21.7141 −1.10522
$$387$$ 26.4524 1.34465
$$388$$ 33.8672 1.71935
$$389$$ −22.1076 −1.12090 −0.560450 0.828188i $$-0.689372\pi$$
−0.560450 + 0.828188i $$0.689372\pi$$
$$390$$ 7.94656 0.402389
$$391$$ −21.7352 −1.09920
$$392$$ −4.25784 −0.215053
$$393$$ −11.8859 −0.599562
$$394$$ 34.8758 1.75702
$$395$$ −0.565815 −0.0284692
$$396$$ −10.2227 −0.513709
$$397$$ −14.6934 −0.737442 −0.368721 0.929540i $$-0.620204\pi$$
−0.368721 + 0.929540i $$0.620204\pi$$
$$398$$ −6.66622 −0.334147
$$399$$ −2.34152 −0.117223
$$400$$ −2.26338 −0.113169
$$401$$ −28.6107 −1.42875 −0.714376 0.699762i $$-0.753289\pi$$
−0.714376 + 0.699762i $$0.753289\pi$$
$$402$$ −11.6133 −0.579218
$$403$$ −35.6136 −1.77404
$$404$$ −22.3987 −1.11438
$$405$$ −4.96265 −0.246596
$$406$$ 0 0
$$407$$ 8.72030 0.432249
$$408$$ 4.31020 0.213387
$$409$$ −6.58740 −0.325726 −0.162863 0.986649i $$-0.552073\pi$$
−0.162863 + 0.986649i $$0.552073\pi$$
$$410$$ −8.86337 −0.437731
$$411$$ −11.9714 −0.590507
$$412$$ 45.7605 2.25446
$$413$$ −6.84979 −0.337056
$$414$$ −26.6968 −1.31208
$$415$$ −10.3651 −0.508803
$$416$$ −41.2495 −2.02242
$$417$$ 13.8917 0.680280
$$418$$ −3.53594 −0.172948
$$419$$ −28.9510 −1.41435 −0.707175 0.707038i $$-0.750031\pi$$
−0.707175 + 0.707038i $$0.750031\pi$$
$$420$$ 5.78052 0.282060
$$421$$ 7.27610 0.354615 0.177308 0.984155i $$-0.443261\pi$$
0.177308 + 0.984155i $$0.443261\pi$$
$$422$$ 42.8062 2.08378
$$423$$ −0.439218 −0.0213555
$$424$$ −10.1892 −0.494830
$$425$$ 4.43761 0.215256
$$426$$ 5.89303 0.285518
$$427$$ −30.3293 −1.46774
$$428$$ 14.6998 0.710544
$$429$$ 5.61505 0.271097
$$430$$ −22.5877 −1.08928
$$431$$ −14.3124 −0.689402 −0.344701 0.938713i $$-0.612020\pi$$
−0.344701 + 0.938713i $$0.612020\pi$$
$$432$$ −8.60735 −0.414121
$$433$$ 6.81929 0.327714 0.163857 0.986484i $$-0.447606\pi$$
0.163857 + 0.986484i $$0.447606\pi$$
$$434$$ −45.4266 −2.18055
$$435$$ 0 0
$$436$$ −36.7804 −1.76146
$$437$$ −5.26615 −0.251914
$$438$$ −19.2273 −0.918718
$$439$$ 18.2642 0.871702 0.435851 0.900019i $$-0.356448\pi$$
0.435851 + 0.900019i $$0.356448\pi$$
$$440$$ 2.15172 0.102579
$$441$$ −7.62117 −0.362913
$$442$$ 51.2466 2.43755
$$443$$ −20.2619 −0.962670 −0.481335 0.876537i $$-0.659848\pi$$
−0.481335 + 0.876537i $$0.659848\pi$$
$$444$$ −10.4481 −0.495847
$$445$$ 3.54640 0.168115
$$446$$ 22.8130 1.08023
$$447$$ 7.52951 0.356134
$$448$$ −38.2887 −1.80897
$$449$$ 2.05073 0.0967798 0.0483899 0.998829i $$-0.484591\pi$$
0.0483899 + 0.998829i $$0.484591\pi$$
$$450$$ 5.45060 0.256944
$$451$$ −6.26288 −0.294907
$$452$$ −16.8860 −0.794249
$$453$$ −7.78354 −0.365703
$$454$$ −12.8618 −0.603636
$$455$$ 16.9413 0.794223
$$456$$ 1.04430 0.0489039
$$457$$ −21.2682 −0.994885 −0.497442 0.867497i $$-0.665727\pi$$
−0.497442 + 0.867497i $$0.665727\pi$$
$$458$$ 5.17541 0.241831
$$459$$ 16.8757 0.787689
$$460$$ 13.0005 0.606153
$$461$$ 21.7537 1.01317 0.506586 0.862190i $$-0.330907\pi$$
0.506586 + 0.862190i $$0.330907\pi$$
$$462$$ 7.16222 0.333217
$$463$$ −22.2978 −1.03627 −0.518134 0.855300i $$-0.673373\pi$$
−0.518134 + 0.855300i $$0.673373\pi$$
$$464$$ 0 0
$$465$$ 4.57814 0.212306
$$466$$ −30.2511 −1.40136
$$467$$ 8.99459 0.416220 0.208110 0.978105i $$-0.433269\pi$$
0.208110 + 0.978105i $$0.433269\pi$$
$$468$$ 35.8967 1.65932
$$469$$ −24.7585 −1.14324
$$470$$ 0.375049 0.0172997
$$471$$ −2.93269 −0.135131
$$472$$ 3.05496 0.140616
$$473$$ −15.9605 −0.733867
$$474$$ 0.839969 0.0385810
$$475$$ 1.07517 0.0493322
$$476$$ 37.2780 1.70863
$$477$$ −18.2377 −0.835049
$$478$$ 22.9218 1.04842
$$479$$ 4.99651 0.228296 0.114148 0.993464i $$-0.463586\pi$$
0.114148 + 0.993464i $$0.463586\pi$$
$$480$$ 5.30263 0.242031
$$481$$ −30.6211 −1.39620
$$482$$ 47.7553 2.17519
$$483$$ 10.6669 0.485359
$$484$$ −23.0290 −1.04677
$$485$$ −12.7595 −0.579379
$$486$$ 31.9799 1.45064
$$487$$ −22.2387 −1.00773 −0.503866 0.863782i $$-0.668089\pi$$
−0.503866 + 0.863782i $$0.668089\pi$$
$$488$$ 13.5267 0.612323
$$489$$ −14.7680 −0.667834
$$490$$ 6.50773 0.293989
$$491$$ −33.1871 −1.49771 −0.748855 0.662733i $$-0.769396\pi$$
−0.748855 + 0.662733i $$0.769396\pi$$
$$492$$ 7.50381 0.338298
$$493$$ 0 0
$$494$$ 12.4163 0.558637
$$495$$ 3.85140 0.173108
$$496$$ −15.0586 −0.676149
$$497$$ 12.5634 0.563546
$$498$$ 15.3873 0.689522
$$499$$ 21.2379 0.950737 0.475369 0.879787i $$-0.342315\pi$$
0.475369 + 0.879787i $$0.342315\pi$$
$$500$$ −2.65427 −0.118703
$$501$$ −8.59338 −0.383924
$$502$$ −19.1448 −0.854475
$$503$$ −23.0259 −1.02667 −0.513337 0.858187i $$-0.671591\pi$$
−0.513337 + 0.858187i $$0.671591\pi$$
$$504$$ 11.2866 0.502744
$$505$$ 8.43874 0.375519
$$506$$ 16.1080 0.716089
$$507$$ −10.7716 −0.478382
$$508$$ −4.16892 −0.184966
$$509$$ 18.7815 0.832477 0.416239 0.909255i $$-0.363348\pi$$
0.416239 + 0.909255i $$0.363348\pi$$
$$510$$ −6.58776 −0.291711
$$511$$ −40.9910 −1.81333
$$512$$ −23.8311 −1.05320
$$513$$ 4.08875 0.180523
$$514$$ 43.1723 1.90425
$$515$$ −17.2403 −0.759699
$$516$$ 19.1230 0.841842
$$517$$ 0.265010 0.0116551
$$518$$ −39.0584 −1.71613
$$519$$ 8.01116 0.351651
$$520$$ −7.55572 −0.331340
$$521$$ −10.8789 −0.476614 −0.238307 0.971190i $$-0.576592\pi$$
−0.238307 + 0.971190i $$0.576592\pi$$
$$522$$ 0 0
$$523$$ −22.1421 −0.968205 −0.484103 0.875011i $$-0.660854\pi$$
−0.484103 + 0.875011i $$0.660854\pi$$
$$524$$ 45.8473 2.00285
$$525$$ −2.17781 −0.0950476
$$526$$ −41.4894 −1.80902
$$527$$ 29.5240 1.28609
$$528$$ 2.37422 0.103325
$$529$$ 0.990052 0.0430458
$$530$$ 15.5732 0.676458
$$531$$ 5.46811 0.237296
$$532$$ 9.03194 0.391584
$$533$$ 21.9919 0.952575
$$534$$ −5.26473 −0.227827
$$535$$ −5.53818 −0.239436
$$536$$ 11.0421 0.476946
$$537$$ −12.5295 −0.540687
$$538$$ 48.1057 2.07398
$$539$$ 4.59838 0.198066
$$540$$ −10.0939 −0.434372
$$541$$ −12.5614 −0.540056 −0.270028 0.962852i $$-0.587033\pi$$
−0.270028 + 0.962852i $$0.587033\pi$$
$$542$$ 10.9948 0.472268
$$543$$ 14.1782 0.608445
$$544$$ 34.1961 1.46615
$$545$$ 13.8570 0.593570
$$546$$ −25.1499 −1.07632
$$547$$ −16.5381 −0.707117 −0.353558 0.935412i $$-0.615028\pi$$
−0.353558 + 0.935412i $$0.615028\pi$$
$$548$$ 46.1773 1.97260
$$549$$ 24.2116 1.03332
$$550$$ −3.28872 −0.140231
$$551$$ 0 0
$$552$$ −4.75734 −0.202486
$$553$$ 1.79074 0.0761500
$$554$$ 51.6448 2.19418
$$555$$ 3.93635 0.167089
$$556$$ −53.5844 −2.27249
$$557$$ 31.2772 1.32526 0.662629 0.748948i $$-0.269441\pi$$
0.662629 + 0.748948i $$0.269441\pi$$
$$558$$ 36.2636 1.53516
$$559$$ 56.0450 2.37045
$$560$$ 7.16333 0.302706
$$561$$ −4.65492 −0.196531
$$562$$ 2.29076 0.0966300
$$563$$ −42.4398 −1.78862 −0.894311 0.447445i $$-0.852334\pi$$
−0.894311 + 0.447445i $$0.852334\pi$$
$$564$$ −0.317520 −0.0133700
$$565$$ 6.36180 0.267643
$$566$$ −19.5678 −0.822495
$$567$$ 15.7062 0.659599
$$568$$ −5.60319 −0.235105
$$569$$ −30.5448 −1.28050 −0.640252 0.768165i $$-0.721170\pi$$
−0.640252 + 0.768165i $$0.721170\pi$$
$$570$$ −1.59612 −0.0668542
$$571$$ 16.0081 0.669917 0.334959 0.942233i $$-0.391278\pi$$
0.334959 + 0.942233i $$0.391278\pi$$
$$572$$ −21.6589 −0.905605
$$573$$ 9.94228 0.415345
$$574$$ 28.0516 1.17085
$$575$$ −4.89796 −0.204259
$$576$$ 30.5654 1.27356
$$577$$ 19.0343 0.792407 0.396204 0.918163i $$-0.370327\pi$$
0.396204 + 0.918163i $$0.370327\pi$$
$$578$$ −5.80841 −0.241598
$$579$$ −6.92594 −0.287832
$$580$$ 0 0
$$581$$ 32.8044 1.36096
$$582$$ 18.9418 0.785165
$$583$$ 11.0041 0.455743
$$584$$ 18.2817 0.756501
$$585$$ −13.5241 −0.559152
$$586$$ −32.6633 −1.34931
$$587$$ 12.4858 0.515345 0.257672 0.966232i $$-0.417044\pi$$
0.257672 + 0.966232i $$0.417044\pi$$
$$588$$ −5.50950 −0.227208
$$589$$ 7.15326 0.294745
$$590$$ −4.66923 −0.192229
$$591$$ 11.1240 0.457580
$$592$$ −12.9475 −0.532141
$$593$$ 28.4529 1.16842 0.584211 0.811602i $$-0.301404\pi$$
0.584211 + 0.811602i $$0.301404\pi$$
$$594$$ −12.5066 −0.513152
$$595$$ −14.0445 −0.575769
$$596$$ −29.0436 −1.18967
$$597$$ −2.12626 −0.0870220
$$598$$ −56.5629 −2.31303
$$599$$ −18.8861 −0.771666 −0.385833 0.922569i $$-0.626086\pi$$
−0.385833 + 0.922569i $$0.626086\pi$$
$$600$$ 0.971290 0.0396527
$$601$$ −36.4020 −1.48487 −0.742434 0.669919i $$-0.766329\pi$$
−0.742434 + 0.669919i $$0.766329\pi$$
$$602$$ 71.4876 2.91362
$$603$$ 19.7644 0.804870
$$604$$ 30.0234 1.22163
$$605$$ 8.67618 0.352737
$$606$$ −12.5276 −0.508897
$$607$$ −31.3635 −1.27300 −0.636502 0.771275i $$-0.719619\pi$$
−0.636502 + 0.771275i $$0.719619\pi$$
$$608$$ 8.28525 0.336011
$$609$$ 0 0
$$610$$ −20.6743 −0.837077
$$611$$ −0.930576 −0.0376471
$$612$$ −29.7586 −1.20292
$$613$$ −21.2613 −0.858735 −0.429368 0.903130i $$-0.641264\pi$$
−0.429368 + 0.903130i $$0.641264\pi$$
$$614$$ 49.5192 1.99843
$$615$$ −2.82707 −0.113998
$$616$$ −6.80996 −0.274381
$$617$$ 1.99430 0.0802875 0.0401437 0.999194i $$-0.487218\pi$$
0.0401437 + 0.999194i $$0.487218\pi$$
$$618$$ 25.5938 1.02953
$$619$$ −17.3604 −0.697774 −0.348887 0.937165i $$-0.613440\pi$$
−0.348887 + 0.937165i $$0.613440\pi$$
$$620$$ −17.6592 −0.709212
$$621$$ −18.6264 −0.747450
$$622$$ 29.7560 1.19311
$$623$$ −11.2239 −0.449678
$$624$$ −8.33700 −0.333747
$$625$$ 1.00000 0.0400000
$$626$$ 55.6680 2.22494
$$627$$ −1.12782 −0.0450409
$$628$$ 11.3122 0.451407
$$629$$ 25.3851 1.01217
$$630$$ −17.2505 −0.687277
$$631$$ −10.6154 −0.422592 −0.211296 0.977422i $$-0.567768\pi$$
−0.211296 + 0.977422i $$0.567768\pi$$
$$632$$ −0.798656 −0.0317688
$$633$$ 13.6535 0.542678
$$634$$ −44.2062 −1.75565
$$635$$ 1.57064 0.0623291
$$636$$ −13.1844 −0.522797
$$637$$ −16.1471 −0.639770
$$638$$ 0 0
$$639$$ −10.0292 −0.396750
$$640$$ −10.6879 −0.422476
$$641$$ 27.5463 1.08801 0.544007 0.839081i $$-0.316907\pi$$
0.544007 + 0.839081i $$0.316907\pi$$
$$642$$ 8.22159 0.324480
$$643$$ 6.19444 0.244285 0.122142 0.992513i $$-0.461024\pi$$
0.122142 + 0.992513i $$0.461024\pi$$
$$644$$ −41.1452 −1.62135
$$645$$ −7.20460 −0.283681
$$646$$ −10.2932 −0.404982
$$647$$ 28.3882 1.11605 0.558027 0.829823i $$-0.311558\pi$$
0.558027 + 0.829823i $$0.311558\pi$$
$$648$$ −7.00486 −0.275177
$$649$$ −3.29929 −0.129508
$$650$$ 11.5482 0.452959
$$651$$ −14.4893 −0.567880
$$652$$ 56.9647 2.23091
$$653$$ −23.7206 −0.928258 −0.464129 0.885767i $$-0.653633\pi$$
−0.464129 + 0.885767i $$0.653633\pi$$
$$654$$ −20.5712 −0.804397
$$655$$ −17.2730 −0.674912
$$656$$ 9.29886 0.363060
$$657$$ 32.7227 1.27663
$$658$$ −1.18699 −0.0462736
$$659$$ −43.0177 −1.67573 −0.837867 0.545875i $$-0.816197\pi$$
−0.837867 + 0.545875i $$0.816197\pi$$
$$660$$ 2.78426 0.108377
$$661$$ −33.6646 −1.30940 −0.654701 0.755888i $$-0.727206\pi$$
−0.654701 + 0.755888i $$0.727206\pi$$
$$662$$ −58.9856 −2.29254
$$663$$ 16.3456 0.634812
$$664$$ −14.6305 −0.567774
$$665$$ −3.40279 −0.131955
$$666$$ 31.1799 1.20820
$$667$$ 0 0
$$668$$ 33.1472 1.28250
$$669$$ 7.27645 0.281324
$$670$$ −16.8769 −0.652011
$$671$$ −14.6085 −0.563955
$$672$$ −16.7822 −0.647388
$$673$$ 16.2493 0.626365 0.313182 0.949693i $$-0.398605\pi$$
0.313182 + 0.949693i $$0.398605\pi$$
$$674$$ 55.9630 2.15562
$$675$$ 3.80288 0.146373
$$676$$ 41.5491 1.59804
$$677$$ −16.6863 −0.641305 −0.320652 0.947197i $$-0.603902\pi$$
−0.320652 + 0.947197i $$0.603902\pi$$
$$678$$ −9.44428 −0.362706
$$679$$ 40.3823 1.54973
$$680$$ 6.26375 0.240204
$$681$$ −4.10242 −0.157205
$$682$$ −21.8803 −0.837840
$$683$$ −28.9333 −1.10710 −0.553550 0.832816i $$-0.686727\pi$$
−0.553550 + 0.832816i $$0.686727\pi$$
$$684$$ −7.21010 −0.275685
$$685$$ −17.3973 −0.664718
$$686$$ 27.1987 1.03845
$$687$$ 1.65075 0.0629801
$$688$$ 23.6976 0.903461
$$689$$ −38.6405 −1.47209
$$690$$ 7.27117 0.276809
$$691$$ −26.3775 −1.00345 −0.501724 0.865028i $$-0.667301\pi$$
−0.501724 + 0.865028i $$0.667301\pi$$
$$692$$ −30.9014 −1.17470
$$693$$ −12.1893 −0.463031
$$694$$ 63.5537 2.41246
$$695$$ 20.1880 0.765774
$$696$$ 0 0
$$697$$ −18.2315 −0.690566
$$698$$ −21.6004 −0.817586
$$699$$ −9.64891 −0.364955
$$700$$ 8.40047 0.317508
$$701$$ 9.51807 0.359492 0.179746 0.983713i $$-0.442472\pi$$
0.179746 + 0.983713i $$0.442472\pi$$
$$702$$ 43.9166 1.65752
$$703$$ 6.15047 0.231969
$$704$$ −18.4422 −0.695067
$$705$$ 0.119626 0.00450537
$$706$$ −44.3130 −1.66774
$$707$$ −26.7076 −1.00444
$$708$$ 3.95301 0.148563
$$709$$ −49.5944 −1.86256 −0.931278 0.364309i $$-0.881305\pi$$
−0.931278 + 0.364309i $$0.881305\pi$$
$$710$$ 8.56398 0.321400
$$711$$ −1.42953 −0.0536115
$$712$$ 5.00580 0.187600
$$713$$ −32.5868 −1.22039
$$714$$ 20.8495 0.780273
$$715$$ 8.16001 0.305167
$$716$$ 48.3299 1.80617
$$717$$ 7.31114 0.273039
$$718$$ −64.5987 −2.41080
$$719$$ −33.3118 −1.24232 −0.621160 0.783684i $$-0.713338\pi$$
−0.621160 + 0.783684i $$0.713338\pi$$
$$720$$ −5.71841 −0.213112
$$721$$ 54.5637 2.03206
$$722$$ 38.4962 1.43268
$$723$$ 15.2321 0.566486
$$724$$ −54.6895 −2.03252
$$725$$ 0 0
$$726$$ −12.8801 −0.478024
$$727$$ −0.974733 −0.0361508 −0.0180754 0.999837i $$-0.505754\pi$$
−0.0180754 + 0.999837i $$0.505754\pi$$
$$728$$ 23.9130 0.886274
$$729$$ −4.68761 −0.173615
$$730$$ −27.9419 −1.03418
$$731$$ −46.4618 −1.71845
$$732$$ 17.5030 0.646931
$$733$$ −9.33744 −0.344886 −0.172443 0.985019i $$-0.555166\pi$$
−0.172443 + 0.985019i $$0.555166\pi$$
$$734$$ −1.83075 −0.0675741
$$735$$ 2.07571 0.0765637
$$736$$ −37.7437 −1.39125
$$737$$ −11.9252 −0.439272
$$738$$ −22.3933 −0.824307
$$739$$ 7.31530 0.269098 0.134549 0.990907i $$-0.457041\pi$$
0.134549 + 0.990907i $$0.457041\pi$$
$$740$$ −15.1837 −0.558162
$$741$$ 3.96032 0.145486
$$742$$ −49.2875 −1.80940
$$743$$ −31.2266 −1.14559 −0.572796 0.819698i $$-0.694141\pi$$
−0.572796 + 0.819698i $$0.694141\pi$$
$$744$$ 6.46212 0.236913
$$745$$ 10.9422 0.400891
$$746$$ −3.22714 −0.118154
$$747$$ −26.1874 −0.958146
$$748$$ 17.9554 0.656515
$$749$$ 17.5277 0.640448
$$750$$ −1.48453 −0.0542074
$$751$$ −10.8850 −0.397199 −0.198599 0.980081i $$-0.563639\pi$$
−0.198599 + 0.980081i $$0.563639\pi$$
$$752$$ −0.393477 −0.0143486
$$753$$ −6.10644 −0.222531
$$754$$ 0 0
$$755$$ −11.3113 −0.411662
$$756$$ 31.9460 1.16186
$$757$$ 14.4670 0.525811 0.262905 0.964822i $$-0.415319\pi$$
0.262905 + 0.964822i $$0.415319\pi$$
$$758$$ −5.46321 −0.198433
$$759$$ 5.13783 0.186491
$$760$$ 1.51762 0.0550499
$$761$$ −10.5919 −0.383955 −0.191977 0.981399i $$-0.561490\pi$$
−0.191977 + 0.981399i $$0.561490\pi$$
$$762$$ −2.33167 −0.0844675
$$763$$ −43.8559 −1.58769
$$764$$ −38.3503 −1.38746
$$765$$ 11.2116 0.405356
$$766$$ −4.98906 −0.180262
$$767$$ 11.5853 0.418323
$$768$$ −0.783162 −0.0282599
$$769$$ 19.0364 0.686472 0.343236 0.939249i $$-0.388477\pi$$
0.343236 + 0.939249i $$0.388477\pi$$
$$770$$ 10.4084 0.375094
$$771$$ 13.7702 0.495923
$$772$$ 26.7154 0.961508
$$773$$ −5.72929 −0.206068 −0.103034 0.994678i $$-0.532855\pi$$
−0.103034 + 0.994678i $$0.532855\pi$$
$$774$$ −57.0678 −2.05126
$$775$$ 6.65314 0.238988
$$776$$ −18.0102 −0.646529
$$777$$ −12.4581 −0.446932
$$778$$ 47.6944 1.70993
$$779$$ −4.41723 −0.158264
$$780$$ −9.77684 −0.350067
$$781$$ 6.05132 0.216533
$$782$$ 46.8911 1.67682
$$783$$ 0 0
$$784$$ −6.82748 −0.243839
$$785$$ −4.26189 −0.152114
$$786$$ 25.6423 0.914630
$$787$$ 18.3560 0.654322 0.327161 0.944969i $$-0.393908\pi$$
0.327161 + 0.944969i $$0.393908\pi$$
$$788$$ −42.9085 −1.52855
$$789$$ −13.2335 −0.471124
$$790$$ 1.22068 0.0434297
$$791$$ −20.1344 −0.715896
$$792$$ 5.43632 0.193171
$$793$$ 51.2973 1.82162
$$794$$ 31.6993 1.12497
$$795$$ 4.96725 0.176170
$$796$$ 8.20161 0.290698
$$797$$ −8.62490 −0.305510 −0.152755 0.988264i $$-0.548814\pi$$
−0.152755 + 0.988264i $$0.548814\pi$$
$$798$$ 5.05155 0.178823
$$799$$ 0.771455 0.0272921
$$800$$ 7.70599 0.272448
$$801$$ 8.95995 0.316584
$$802$$ 61.7241 2.17955
$$803$$ −19.7438 −0.696744
$$804$$ 14.2881 0.503903
$$805$$ 15.5015 0.546356
$$806$$ 76.8320 2.70629
$$807$$ 15.3438 0.540128
$$808$$ 11.9114 0.419042
$$809$$ 7.98842 0.280858 0.140429 0.990091i $$-0.455152\pi$$
0.140429 + 0.990091i $$0.455152\pi$$
$$810$$ 10.7063 0.376181
$$811$$ 11.0531 0.388128 0.194064 0.980989i $$-0.437833\pi$$
0.194064 + 0.980989i $$0.437833\pi$$
$$812$$ 0 0
$$813$$ 3.50692 0.122993
$$814$$ −18.8130 −0.659394
$$815$$ −21.4615 −0.751763
$$816$$ 6.91144 0.241949
$$817$$ −11.2570 −0.393834
$$818$$ 14.2115 0.496893
$$819$$ 42.8022 1.49563
$$820$$ 10.9048 0.380813
$$821$$ 43.4087 1.51497 0.757487 0.652851i $$-0.226427\pi$$
0.757487 + 0.652851i $$0.226427\pi$$
$$822$$ 25.8269 0.900815
$$823$$ −0.554965 −0.0193449 −0.00967243 0.999953i $$-0.503079\pi$$
−0.00967243 + 0.999953i $$0.503079\pi$$
$$824$$ −24.3350 −0.847749
$$825$$ −1.04897 −0.0365205
$$826$$ 14.7776 0.514178
$$827$$ 28.1254 0.978016 0.489008 0.872279i $$-0.337359\pi$$
0.489008 + 0.872279i $$0.337359\pi$$
$$828$$ 32.8458 1.14147
$$829$$ −18.3590 −0.637635 −0.318818 0.947816i $$-0.603286\pi$$
−0.318818 + 0.947816i $$0.603286\pi$$
$$830$$ 22.3614 0.776177
$$831$$ 16.4726 0.571430
$$832$$ 64.7593 2.24513
$$833$$ 13.3860 0.463799
$$834$$ −29.9697 −1.03776
$$835$$ −12.4882 −0.432173
$$836$$ 4.35035 0.150460
$$837$$ 25.3011 0.874533
$$838$$ 62.4583 2.15759
$$839$$ −3.81375 −0.131665 −0.0658327 0.997831i $$-0.520970\pi$$
−0.0658327 + 0.997831i $$0.520970\pi$$
$$840$$ −3.07402 −0.106064
$$841$$ 0 0
$$842$$ −15.6973 −0.540964
$$843$$ 0.730663 0.0251654
$$844$$ −52.6656 −1.81282
$$845$$ −15.6537 −0.538502
$$846$$ 0.947559 0.0325777
$$847$$ −27.4591 −0.943507
$$848$$ −16.3384 −0.561063
$$849$$ −6.24135 −0.214203
$$850$$ −9.57359 −0.328371
$$851$$ −28.0186 −0.960465
$$852$$ −7.25034 −0.248392
$$853$$ 3.39403 0.116209 0.0581047 0.998310i $$-0.481494\pi$$
0.0581047 + 0.998310i $$0.481494\pi$$
$$854$$ 65.4318 2.23903
$$855$$ 2.71641 0.0928993
$$856$$ −7.81723 −0.267187
$$857$$ −7.11361 −0.242996 −0.121498 0.992592i $$-0.538770\pi$$
−0.121498 + 0.992592i $$0.538770\pi$$
$$858$$ −12.1138 −0.413558
$$859$$ 39.4010 1.34434 0.672172 0.740395i $$-0.265362\pi$$
0.672172 + 0.740395i $$0.265362\pi$$
$$860$$ 27.7903 0.947640
$$861$$ 8.94734 0.304925
$$862$$ 30.8771 1.05168
$$863$$ 3.05637 0.104040 0.0520200 0.998646i $$-0.483434\pi$$
0.0520200 + 0.998646i $$0.483434\pi$$
$$864$$ 29.3049 0.996975
$$865$$ 11.6421 0.395845
$$866$$ −14.7118 −0.499927
$$867$$ −1.85265 −0.0629195
$$868$$ 55.8895 1.89701
$$869$$ 0.862532 0.0292594
$$870$$ 0 0
$$871$$ 41.8751 1.41888
$$872$$ 19.5594 0.662365
$$873$$ −32.2368 −1.09105
$$874$$ 11.3611 0.384294
$$875$$ −3.16489 −0.106993
$$876$$ 23.6559 0.799258
$$877$$ −5.14366 −0.173689 −0.0868446 0.996222i $$-0.527678\pi$$
−0.0868446 + 0.996222i $$0.527678\pi$$
$$878$$ −39.4027 −1.32978
$$879$$ −10.4183 −0.351400
$$880$$ 3.45031 0.116310
$$881$$ −17.1521 −0.577870 −0.288935 0.957349i $$-0.593301\pi$$
−0.288935 + 0.957349i $$0.593301\pi$$
$$882$$ 16.4417 0.553622
$$883$$ 19.8973 0.669597 0.334799 0.942290i $$-0.391332\pi$$
0.334799 + 0.942290i $$0.391332\pi$$
$$884$$ −63.0499 −2.12060
$$885$$ −1.48930 −0.0500623
$$886$$ 43.7125 1.46855
$$887$$ −9.25219 −0.310658 −0.155329 0.987863i $$-0.549644\pi$$
−0.155329 + 0.987863i $$0.549644\pi$$
$$888$$ 5.55622 0.186454
$$889$$ −4.97091 −0.166719
$$890$$ −7.65092 −0.256459
$$891$$ 7.56510 0.253440
$$892$$ −28.0674 −0.939766
$$893$$ 0.186913 0.00625481
$$894$$ −16.2440 −0.543281
$$895$$ −18.2083 −0.608637
$$896$$ 33.8260 1.13005
$$897$$ −18.0413 −0.602382
$$898$$ −4.42419 −0.147637
$$899$$ 0 0
$$900$$ −6.70601 −0.223534
$$901$$ 32.0333 1.06718
$$902$$ 13.5114 0.449880
$$903$$ 22.8017 0.758794
$$904$$ 8.97978 0.298663
$$905$$ 20.6043 0.684911
$$906$$ 16.7920 0.557878
$$907$$ −1.08515 −0.0360317 −0.0180159 0.999838i $$-0.505735\pi$$
−0.0180159 + 0.999838i $$0.505735\pi$$
$$908$$ 15.8242 0.525146
$$909$$ 21.3204 0.707154
$$910$$ −36.5489 −1.21158
$$911$$ 28.9580 0.959423 0.479711 0.877426i $$-0.340741\pi$$
0.479711 + 0.877426i $$0.340741\pi$$
$$912$$ 1.67455 0.0554498
$$913$$ 15.8006 0.522925
$$914$$ 45.8835 1.51769
$$915$$ −6.59428 −0.218000
$$916$$ −6.36743 −0.210386
$$917$$ 54.6670 1.80526
$$918$$ −36.4072 −1.20162
$$919$$ 5.75933 0.189983 0.0949915 0.995478i $$-0.469718\pi$$
0.0949915 + 0.995478i $$0.469718\pi$$
$$920$$ −6.91355 −0.227933
$$921$$ 15.7947 0.520452
$$922$$ −46.9310 −1.54559
$$923$$ −21.2490 −0.699421
$$924$$ −8.81186 −0.289889
$$925$$ 5.72046 0.188087
$$926$$ 48.1048 1.58082
$$927$$ −43.5576 −1.43062
$$928$$ 0 0
$$929$$ 18.7476 0.615088 0.307544 0.951534i $$-0.400493\pi$$
0.307544 + 0.951534i $$0.400493\pi$$
$$930$$ −9.87678 −0.323872
$$931$$ 3.24325 0.106293
$$932$$ 37.2187 1.21914
$$933$$ 9.49098 0.310721
$$934$$ −19.4047 −0.634942
$$935$$ −6.76472 −0.221230
$$936$$ −19.0895 −0.623959
$$937$$ 19.5861 0.639851 0.319926 0.947443i $$-0.396342\pi$$
0.319926 + 0.947443i $$0.396342\pi$$
$$938$$ 53.4134 1.74401
$$939$$ 17.7559 0.579442
$$940$$ −0.461432 −0.0150503
$$941$$ 44.5990 1.45388 0.726942 0.686698i $$-0.240941\pi$$
0.726942 + 0.686698i $$0.240941\pi$$
$$942$$ 6.32691 0.206142
$$943$$ 20.1228 0.655289
$$944$$ 4.89865 0.159437
$$945$$ −12.0357 −0.391521
$$946$$ 34.4329 1.11951
$$947$$ −2.83282 −0.0920544 −0.0460272 0.998940i $$-0.514656\pi$$
−0.0460272 + 0.998940i $$0.514656\pi$$
$$948$$ −1.03343 −0.0335644
$$949$$ 69.3299 2.25054
$$950$$ −2.31955 −0.0752561
$$951$$ −14.1000 −0.457225
$$952$$ −19.8241 −0.642501
$$953$$ 20.6932 0.670319 0.335160 0.942161i $$-0.391210\pi$$
0.335160 + 0.942161i $$0.391210\pi$$
$$954$$ 39.3457 1.27386
$$955$$ 14.4485 0.467543
$$956$$ −28.2012 −0.912093
$$957$$ 0 0
$$958$$ −10.7794 −0.348265
$$959$$ 55.0606 1.77800
$$960$$ −8.32483 −0.268683
$$961$$ 13.2642 0.427878
$$962$$ 66.0612 2.12990
$$963$$ −13.9922 −0.450892
$$964$$ −58.7545 −1.89236
$$965$$ −10.0650 −0.324005
$$966$$ −23.0124 −0.740413
$$967$$ 8.62483 0.277356 0.138678 0.990338i $$-0.455715\pi$$
0.138678 + 0.990338i $$0.455715\pi$$
$$968$$ 12.2466 0.393620
$$969$$ −3.28314 −0.105470
$$970$$ 27.5270 0.883840
$$971$$ 6.65773 0.213657 0.106828 0.994277i $$-0.465930\pi$$
0.106828 + 0.994277i $$0.465930\pi$$
$$972$$ −39.3457 −1.26201
$$973$$ −63.8926 −2.04830
$$974$$ 47.9773 1.53729
$$975$$ 3.68343 0.117964
$$976$$ 21.6901 0.694283
$$977$$ 18.4264 0.589512 0.294756 0.955573i $$-0.404762\pi$$
0.294756 + 0.955573i $$0.404762\pi$$
$$978$$ 31.8602 1.01878
$$979$$ −5.40615 −0.172781
$$980$$ −8.00662 −0.255762
$$981$$ 35.0097 1.11777
$$982$$ 71.5970 2.28475
$$983$$ −1.45926 −0.0465431 −0.0232715 0.999729i $$-0.507408\pi$$
−0.0232715 + 0.999729i $$0.507408\pi$$
$$984$$ −3.99045 −0.127211
$$985$$ 16.1658 0.515086
$$986$$ 0 0
$$987$$ −0.378602 −0.0120510
$$988$$ −15.2761 −0.485998
$$989$$ 51.2817 1.63066
$$990$$ −8.30893 −0.264075
$$991$$ 43.4928 1.38159 0.690796 0.723049i $$-0.257260\pi$$
0.690796 + 0.723049i $$0.257260\pi$$
$$992$$ 51.2690 1.62779
$$993$$ −18.8141 −0.597047
$$994$$ −27.1040 −0.859687
$$995$$ −3.08996 −0.0979585
$$996$$ −18.9314 −0.599864
$$997$$ −15.2215 −0.482071 −0.241035 0.970516i $$-0.577487\pi$$
−0.241035 + 0.970516i $$0.577487\pi$$
$$998$$ −45.8181 −1.45035
$$999$$ 21.7542 0.688273
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4205.2.a.r.1.1 12
29.4 even 14 145.2.k.a.16.1 24
29.22 even 14 145.2.k.a.136.1 yes 24
29.28 even 2 4205.2.a.q.1.12 12
145.4 even 14 725.2.l.d.451.4 24
145.22 odd 28 725.2.r.c.49.1 48
145.33 odd 28 725.2.r.c.74.1 48
145.62 odd 28 725.2.r.c.74.8 48
145.109 even 14 725.2.l.d.426.4 24
145.138 odd 28 725.2.r.c.49.8 48

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.k.a.16.1 24 29.4 even 14
145.2.k.a.136.1 yes 24 29.22 even 14
725.2.l.d.426.4 24 145.109 even 14
725.2.l.d.451.4 24 145.4 even 14
725.2.r.c.49.1 48 145.22 odd 28
725.2.r.c.49.8 48 145.138 odd 28
725.2.r.c.74.1 48 145.33 odd 28
725.2.r.c.74.8 48 145.62 odd 28
4205.2.a.q.1.12 12 29.28 even 2
4205.2.a.r.1.1 12 1.1 even 1 trivial